# 8.1: What's the Value

Difficulty Level: At Grade Created by: CK-12

Look at the equation below. Can you figure out the value of \begin{align*}z\end{align*}? In this section, we will learn how to use the order of operations to help us to solve equations.

\begin{align*}6 \times 3^2 \div 2 + z + 4(7 - 3) + 2^3 \div 4 = 3^2 \times 6 + 1\end{align*}

### What's the Value

The order of operations tells us the correct order of evaluating math expressions. We always do parentheses first and then exponents. Next we do multiplication and division (from left to right) and finally addition and subtraction (from left to right).

In order to evaluate expressions using the order of operations, we can use the problem solving steps to help.

• First, describe what you see in the problem. What operations are there?
• Second, identify what your job is. In these problems, your job will be to solve for the unknown.
• Third, make a plan. In these problems, your plan should be to use the order of operations.
• Fourth, solve.
• Fifth, check. Substitute your answer into the equation and make sure it works.

#### Finding Unknown Values

1. Follow the order of operations and show each step. What is the value of the variable?

\begin{align*} b + 2 \times 3 \times 2^2 \div 3 = 2(5 + 6) - 2\end{align*}

We can use the problem solving steps to help us with the order of operations.

\begin{align*}& \mathbf{Describe:} && \text{The equation has parentheses, exponents, multiplication, division, subtraction and addition.}\\ &&& b \ \text{is the variable.}\\ \\ & \mathbf{My \ Job:} && \text{Apply the order of operations rule to figure out the value of}\ b.\\ \\ & \mathbf{Solve:} && b + 2 \times 3 \times 2^2 \div 3 = 2(5 + 6) - 2\\ &&& \mathbf{Parentheses} \qquad \qquad \quad b + 2 \times 3 \times 2^2 \div 3 = 2 \times 11 - 2\\ &&& \mathbf{Exponents} \qquad \qquad \quad \ b + 2 \times 3 \times 4 \div 3 = 2 \times 11 - 2\\ &&& \mathbf{Multiplication/} \qquad \quad b + 8 = 22 - 2\\ &&& \mathbf{Division}\\ &&& \mathbf{(left \ to \ right)}\\ &&& \mathbf{Addition/} \qquad \qquad \qquad \ b + 8 = 20\\ &&& \mathbf{Subtraction}\\ &&& \mathbf{(left \ to \ right)} \qquad \qquad \ b=20-8\\ &&& \qquad \qquad \qquad \qquad \qquad \quad b=12\\ \\ & \mathbf{Check:} && \text{Replace} \ b \ \text{with 12 in the equation. Check that the two expressions}\\ &&&\text{(to the right and to the left of the = symbol) name the same number.}\\ &&& 12 + 2 \times 3 \times 2^2 \div 3 = 2(5 + 6) - 2\\ &&& 12 + 2 \times 3 \times 2^2 \div 3 = 2\times 11 - 2\\ &&& 12 + 2 \times 3 \times 4 \div 3 = 2\times 11 - 2\\ &&& 12 + 8 = 22 - 2\\ &&& 20 = 20\end{align*}

2. Follow the order of operations and show each step. What is the value of the variable?

\begin{align*}10^2 - 6(4 + 6) - 3^2 - 3 \times 4 - 1 = d + 50 \div 5^2\end{align*}

We can use the problem solving steps to help us with the order of operations.

\begin{align*}& \mathbf{Describe:} && \text{The equation has parentheses, exponents, multiplication, subtraction, and addition.}\\ &&& d \ \text{is the variable.}\\ \\ & \mathbf{My \ Job:} && \text{Apply the order of operations rule to figure out the value of}\ d.\\ \\ & \mathbf{Solve:} && 10^2 - 6(4 + 6) - 3^2 - 3 \times 4 - 1 = d + 50 \div 5^2\\ &&& \mathbf{Parentheses} \qquad \qquad \quad 10^2 - 6\times 10 - 3^2 - 3 \times 4 - 1 = d + 50 \div 5^2\\ &&& \mathbf{Exponents} \qquad \qquad \quad \ 100 - 6\times 10 - 9 - 3 \times 4 - 1 = d + 50 \div 25\\ &&& \mathbf{Multiplication/} \qquad \quad 100 - 60 - 9 - 12 - 1 = d + 2\\ &&& \mathbf{Division}\\ &&& \mathbf{(left \ to \ right)}\\ &&& \mathbf{Addition/} \qquad \qquad \qquad \ 18=d+2\\ &&& \mathbf{Subtraction}\\ &&& \mathbf{(left \ to \ right)} \qquad \qquad \ 18 -2= d\\ &&& \qquad \qquad \qquad \qquad \qquad \quad d = 16\\ \\ & \mathbf{Check:} && \text{Replace} \ d \ \text{with 16 in the equation. Check that the two expressions}\\ &&&\text{(to the right and to the left of the = symbol) name the same number.}\\ &&& 10^2 - 6(4 + 6) - 3^2 - 3 \times 4 - 1 = 16 + 50 \div 5^2\\ &&& 10^2 - 6\times 10 - 3^2 - 3 \times 4 - 1 = 16 + 50 \div 5^2\\ &&& 100 - 6\times 10 - 9 - 3 \times 4 - 1 = 16 + 50 \div 25\\ &&& 100 - 60 - 9 - 12 - 1 = 16 + 2\\ &&& 18 = 18\end{align*}

3. Follow the order of operations and show each step. What is the value of the variable?

\begin{align*}4(9 - 5) + h + 3^2 - 2^3 + 4^1 = 3^2 \times 3 - 2 \times 3\end{align*}

We can use the problem solving steps to help us with the order of operations.

\begin{align*}& \mathbf{Describe:} && \text{The equation has parentheses, exponents, multiplication, subtraction, and addition.}\\ &&& h \ \text{is the variable.}\\ \\ & \mathbf{My \ Job:} && \text{Apply the order of operations rule to figure out the value of}\ h.\\ \\ & \mathbf{Solve:} && 4(9 - 5) + h + 3^2 - 2^3 + 4^1 = 3^2 \times 3 - 2 \times 3\\ &&& \mathbf{Parentheses} \qquad \qquad \quad 4 \times 4 + h + 3^2 - 2^3 + 4^1 = 3^2 \times 3 - 2 \times 3\\ &&& \mathbf{Exponents} \qquad \qquad \quad \ 4 \times 4 + h + 9 - 8 + 4 = 9 \times 3 - 2 \times 3\\ &&& \mathbf{Multiplication/} \qquad \quad 16 + h + 9 - 8 + 4 = 27 - 6\\ &&& \mathbf{Division}\\ &&& \mathbf{(left \ to \ right)}\\ &&& \mathbf{Addition/} \qquad \qquad \qquad \ h + 21 = 21\\ &&& \mathbf{Subtraction}\\ &&& \mathbf{(left \ to \ right)} \qquad \qquad \ h = 21 - 21\\ &&& \qquad \qquad \qquad \qquad \qquad \quad h = 0\\ \\ & \mathbf{Check:} && \text{Replace} \ h \ \text{with 0 in the equation. Check that the two expressions}\\ &&&\text{(to the right and to the left of the = symbol) name the same number.}\\ &&& 4(9 - 5) + 0 + 3^2 - 2^3 + 4^1 = 3^2 \times 3 - 2 \times 3\\ &&& 4\times 4 + 0 + 3^2 - 2^3 + 4^1 = 3^2 \times 3 - 2 \times 3\\ &&& 4\times 4 + 0 + 9 - 8 + 4 = 9 \times 3 - 2 \times 3\\ &&& 16 + 0 + 9 - 8 + 4 = 27 - 6\\ &&& 21 = 21\end{align*}

#### Earlier Problem Revisited

\begin{align*}6 \times 3^2 \div 2 + z + 4(7 - 3) + 2^3 \div 4 = 3^2 \times 6 + 1\end{align*}

We can use the problem solving steps to help us with the order of operations.

\begin{align*}& \mathbf{Describe:} && \text{The equation has parentheses, exponents, multiplication, division, and addition.}\\ &&& z \ \text{is the variable.}\\ \\ & \mathbf{My \ Job:} && \text{Apply the order of operations rule to figure out the value of}\ z.\\ \\ & \mathbf{Solve:} && 6 \times 3^2 \div 2 + z + 4(7 - 3) + 2^3 \div 4 = 3^2 \times 6 + 1\\ &&& \mathbf{Parentheses} \qquad \qquad \quad 6 \times 3^2 \div 2 + z + 4 \times 4 + 2^3 \div 4 = 3^2 \times 6 + 1\\ &&& \mathbf{Exponents} \qquad \qquad \quad \ 6 \times 9 \div 2 + z + 4 \times 4 + 8 \div 4 = 9 \times 6 + 1\\ &&& \mathbf{Multiplication/} \qquad \quad 27 + z + 16 + 2 = 54 + 1\\ &&& \mathbf{Division}\\ &&& \mathbf{(left \ to \ right)}\\ &&& \mathbf{Addition} \qquad \qquad \qquad \ 45 + z = 55\\ &&& \mathbf{(left \ to \ right)} \qquad \qquad \ z = 55 - 45\\ &&& \qquad \qquad \qquad \qquad \qquad \quad z = 10\\ \\ & \mathbf{Check:} && \text{Replace} \ z \ \text{with 10 in the equation. Check that the two expressions}\\ &&&\text{(to the right and to the left of the = symbol) name the same number.}\\ &&& 6 \times 3^2 \div 2 + 10 + 4(7 - 3) + 2^3 \div 4 = 3^2 \times 6 + 1\\ &&& 6 \times 3^2 \div 2 + 10 + 4 \times 4 + 2^3 \div 4 = 3^2 \times 6 + 1\\ &&& 6 \times 9 \div 2 + 10 + 4 \times 4 + 8 \div 4 = 9 \times 6 + 1\\ &&& 27 + 10 + 16 + 2 = 54 + 1\\ &&& 55 = 55\end{align*}

### Vocabulary

The order of operations tells us the correct order of evaluating math expressions. We always do parentheses first and then exponents. Next we do multiplication and division (from left to right) and finally addition and subtraction (from left to right).

### Examples

For each problem, follow the order of operations and show each step. What is the value of the variable?

#### Example 1

\begin{align*}2 + 5^2 \div 5 \times 1 + 0 \times 34 = 2y - 7(4 - 3)\end{align*}

\begin{align*} 2 + 5^2 \div 5 \times 1 + 0 \times 34 &= 2y - 7(4 - 3)\\ 2 + 5^2 \div 5 \times 1 + 0 \times 34 &= 2y - 7 \times 1\\ 2 + 25 \div 5 \times 1 + 0 \times 34 &= 2y - 7 \times 1\\ 2 + 5 + 0 &= 2y - 7\\ 7 &= 2y - 7\\ 7 + 7 &= 2y\\ 14 &= 2y\\ 7 &= y\end{align*}

#### Example 2

\begin{align*}2a + 5(9 - 8) \times 2^2 = 2^2 \times 3^2 + 2\end{align*}

\begin{align*} 2a + 5(9 - 8) \times 2^2 &= 2^2 \times 3^2 + 2\\ 2a + 5 \times 1 \times 2^2 &= 2^2 \times 3^2 + 2\\ 2a + 5 \times 1 \times 4 &= 4 \times 9 + 2\\ 2a + 20 &= 36 + 2\\ 2a + 20 &= 38\\ 2a &= 38 - 20\\ 2a &= 18\\ a &= 9\end{align*}

#### Example 3

\begin{align*}9^2 - 8^2 - 16 + 4^3 + 2^2 = e(5 + 2) - 1\end{align*}

\begin{align*} 9^2 - 8^2 - 16 + 4^3 + 2^2 &= e(5 + 2) - 1\\ 9^2 - 8^2 - 16 + 4^3 + 2^2 &= e \times 7 - 1\\ 81 - 64 - 16 + 64 + 4&= 7e - 1\\ 69 &= 7e - 1\\ 69 + 1 &= 7e\\ 70 &= 7e\\ 10 &= e\end{align*}

### Review

For each problem, follow the order of operations and show each step. What is the value of the variable?

1. \begin{align*}2m + 3 \times 9 \div 3^3 + 4(8 - 3) = 7^2 - 3 \times 6\end{align*}
2. \begin{align*}7^2 \div 7 \times (5^2 - 17) - (2 \times 6) = 4l - 2^2 \times 5\end{align*}
3. \begin{align*}6+3^2 \div 3 \times 2 +1 \times 5 =3y-2(5-3)\end{align*}
4. \begin{align*}5m+2(7-6)\times 2^3 = 2^2 \times 4^2 +7\end{align*}
5. \begin{align*}3^2-2^2-15+3^3+2^3=f(2+1)-2\end{align*}

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Jan 18, 2013